Open AccessGraph Realizations Constrained by Connected Local Dimensions and Connected Local Bases
Author(s) -
Varanoot Khemmani,
Witsarut Pho-on,
Supachoke Isariyapalakul
Publication year - 2022
Publication title -
wseas transactions on mathematics
Language(s) - English
Resource type - Journals
eISSN - 2224-2880
pISSN - 1109-2769
DOI - 10.37394/23206.2022.21.1
Subject(s) - combinatorics , connectivity , vertex connectivity , mathematics , vertex (graph theory) , connected component , graph , simply connected space , basis (linear algebra) , strongly connected component , cardinality (data modeling) , discrete mathematics , computer science , geometry , data mining
For an ordered set W = {w1,w2, ...,wk} of k distinct vertices in a connected graph G, the representation of a vertex v of G with respect to W is the k-vector r(v|W) = (d(v,w1), d(v,w2), ..., d(v,wk)), where d(v,wi) is the distance from v to wi for 1 ≤ i ≤ k. The setW is called a connected local resolving set of G if the representations of every two adjacent vertices of G with respect to W are distinct and the subgraph ⟨W⟩ induced by W is connected. A connected local resolving set of G of minimum cardinality is a connected local basis of G. The connected local dimension cld(G) of G is the cardinality of a connected local basis of G. In this paper, the connected local dimensions of some well-known graphs are determined. We study the relationship between connected local bases and local bases in a connected graph, and also present some realization results.